A hall forgets almost everything but the last part to leave

Orchestrators do not thin a texture at random. A piece that fades may lose its piccolo before its violas, or its basses before its flutes, and either choice has a long history behind it and a clear effect in the score. A hall leaves a thinning ending most of its diminuendo calculated how much of a written drop survives a room, and it did so with a tail that had no spectrum, a part being simply a part. The essay closed on the obvious objection. A real hall’s high frequencies die sooner than its low ones. So a score that drops its upper parts first leaves the room’s darkness to fill the gap, while a score that drops its bass first takes away exactly what the tail holds longest. On that reasoning the order of leaving should decide how much of a diminuendo reaches the seats, and a thinning from the top down should be the one that sounds as written.

The argument is reasonable, and the computation it asks for is straightforward. The answer is not the one it expects. A tail with a spectrum changes hardly anything. What changes the ending, by more than three phons across all possible orders, is something much plainer, and it was sitting in the earlier calculation’s choice of a single line to draw.

The model is the one the page can write a diminuendo built, on the excitation pattern, with one change. Six string parts, G3, C4, E4, G4, C5 and E5, each at 62 decibels per fundamental, stay together for eight two-second bars and then drop out one a bar until one is left. The excitation pattern is read over direct plus reverberant power, and the reverberant field is now carried partial by partial. Each partial’s field fills and empties at its own reverberation time, which is the hall’s middle-band figure multiplied by a curve: rising to 1.3 times below 125 hertz, falling to 0.55 at 4 kilohertz, a round description of a concert hall with air absorption at the top. A band’s steady field scales with its own reverberation time, as the energy balance says it must, and at middle frequencies it stands six decibels above the direct sound, as in the earlier essay. The short-term and long-term smoothers are the published ones used throughout.

That is the same passage thinned three ways through its closing bars, with the mid-band reverberation on the dial. All three start level, descend in steps, and end near each other, and what distinguishes the three lines is the last bar and the one before it.

Only the note left behind matters, with no room at all #

Before bringing a hall in, it is worth seeing what the order does without one, because that settles most of the question.

With no room, the order of leaving does not matter in the least. The last bar is one note, and at the moment of reading it there is nothing else in the air. Every order that ends on a given note gives the same drop to the second decimal place, as they must, and it is worth stating because it shows where any effect of order has to come from: a room. All that a dry ending shows is the note left sounding, and even that is a substantial effect. Left on G3, the drop from a six-part texture is 10.72 phons. Left on E5, it is 8.47. That is 2.25 phons, and no choice of hall gets near it. The reason is not about reverberation. It is that a low note at a given sound level is quieter than a higher one at the same level, by the equal-loudness relation, and a written thinning that ends on the bass line has therefore gone further down whatever the room does.

So the choice of which part to leave for last was always a choice of how deep the diminuendo would be. A hand-written drop into a solo bass is not the same hairpin as a drop into a solo flute at the same printed dynamic, and the difference is about a fifth of the whole gesture. Nobody writes that into a score, and a page has twelve phons would not have predicted anything else: this is a fact about the excitation pattern at low level.

A hall remembers one note, and it is the one that just left #

Now add the hall. In a 2-second room, the ending’s drop becomes a range for each last note instead of a single value, from 9.39 to 8.05 phons on G3 and from 7.40 to 6.04 on E5. Which earlier order is responsible for that spread? A search over all 720 orders of six parts gives a blunt answer: the note that left just before the last bar is the only earlier note that matters.

The result is sharper than expected. Take E5, the highest and on this pattern one of the loudest, and move it through the order while the ending stays on G3. If E5 goes first, second, third or fourth, the drop is 9.39 phons all four times, unchanged to the second decimal. If it goes last, the drop is 8.05. A two-second hall has nothing left of a note after two bars: at sixty decibels in two seconds, the field is sixty decibels down by the end of the next bar, and it adds nothing to a note sounding at 62. Even a stone-built eight-second hall is fifteen decibels down each bar, thirty after two, and on a pattern where the surviving note is itself raising the threshold, thirty decibels down is under the threshold of mattering. The fourth place starts to show there, by a quarter of a phon. The first four decisions in the order have no effect on the ending anywhere that music is ordinarily played.

What does the last leaver do? It fills the final bar with a fresh tail, which at the moment of reading is only 0.3 seconds old and barely fallen, and that tail is the room’s contribution to the arrival. The dependence on which note it is follows from what the tail needs to do. A last leaver that is low and close to the remaining note is partly under that note’s own excitation and adds little. One that is high and well separated adds a whole new region of the pattern. That is why E5 last on a G3 ending raises the arrival by a phon and a third, while a C4 last hardly raises it. An ending’s drop in a room is a question about a two-note chord that nobody wrote: the last note and the ghost of the one before.

The tail’s darkness moves it by under half a phon #

The argument the earlier essay ended on was that spectrum should matter because the bass outlasts the treble. It does, and the effect is small.

Giving the tail a spectrum moves the top-down ending by a tenth of a phon and the bass-up ending by under half. And the direction is the opposite of what the reasoning predicted. It said that a bass-up thinning would suffer, losing what the tail holds longest, and that a top-down thinning would benefit. In fact the bass-up thinning benefits. Its drop is deeper with a spectrum than without, −7.40 against −6.95, and a darker hall deepens it further to −7.62. The reason is that a bass-up ending’s last leaver is C5, a high note. Its tail is what fills the final bar, and with a spectrum that tail is the part that dies sooner. The earlier reasoning looked at the part the score removes first, on the assumption that the tail held a memory of it. It holds no such memory, as the last section showed. The only part that matters is the last one out, and when that is high the room’s shorter treble helps the diminuendo. A top-down ending’s last leaver is C4, whose tail lies in the middle band, and nothing much happens either way.

The expected mechanism is thus not exactly wrong but mis-aimed. A top-down thinning does leave the room’s darkness to fill the gap, for one bar. The earlier bars have gone. This is a useful correction to how to read an orchestration: the sense of a passage losing its brightness into a dark, warm hall is real and belongs to the last bar, not the whole descent.

Across rooms the two plain orders behave almost identically, and that was the original question.

At two seconds the top-down ending keeps 88 per cent of its dry drop and the bass-up ending 87. The whole concern about thinning order, as a matter of the hall’s survival, is a percentage point. Only one of the three orders behaves differently, and it is the one that was included only as a contrast. “Inner first” drops the middle voices and leaves the outer pair until the very end, so that its last leaver is E5 and its last note is G3. That gives it the deepest possible dry drop and the highest and most separated possible ghost, and so a hall takes more from it than from either plain order: 75 per cent at 2 seconds, 61 at 4, 51 in a cathedral. The order of leaving matters in a room only in that it decides which note haunts the last bar, and the worst case for a composer who wants a clean arrival is a wide spacing that closes from the top at the last moment.

That is the surprising connection. The same quantity has already appeared under another name. A rest needs a dry room found that a rest before an arrival fills with the tail of what preceded it and loses its effect, and a staccato is a dynamic mark worked from the same fact about what a note’s end leaves behind. Thinning was treated separately because it looked like a gradual process with many steps. In a room it is a succession of single-bar events, and each of them is over before the next one matters. A diminuendo of six steps is, as far as any one moment is concerned, a sequence of two-note events.

What a listener hears along the way #

So far the drop has been measured from the steady texture. A listener does not hold that texture in mind indefinitely: loud is relative and it comes down slowly, and the impression has been sinking with the music.

Against that moving reference the picture changes somewhat. Dry, the inner-first order ends by far the deepest under the listener’s impression, −6.73 against −4.47 for the top-down order, even though its written drop is the same to two decimal places: its last step takes away E5, nearly two octaves above the G3 that stays, while the other orders take away a note next door. It arrives with the largest single step in the passage, and a step is what an impression notices. In an 8-second hall that extra depth is entirely gone, −2.41 against −2.56. So the one order that a room strips of most of its drop is also the only one a dry room gives a real arrival. A composer writing for a known dry room could use that wide final step, and in a resonant room it is wasted.

The hall closes the gap between written drop and impression in every case, by a bit under half for the plain orders and by nearly two thirds for the inner-first one. So the plain thinning, whichever way it runs, still reaches the last bar two to four phons under what the listener has come to expect, in any room that music is normally played in. The conclusion from the earlier essay, that a hall leaves a thinning most of its diminuendo, is not disturbed.

What this tail cannot do #

The parts are not instruments. Every part is the same string timbre at the same level. Real parts differ in spectrum, level and directivity, and a flute and a double bass will not line up with a theoretical pair of string tones. A last leaver’s effect, a phon and a third here, depends on how much of the pattern it occupies alone, and on how far it is from the remaining note. Both vary with instrument, and neither variation has been computed. The chord is also one arbitrary six-part spacing of a C major chord, chosen because the earlier essays use it.

It is one number per seat. The field here is a single reverberant level at every seat, 6 decibels over direct, with no distance and no direction. A seat near the stage has a much smaller field and a seat at the back a larger one, and the frequency curve itself varies with where the seat is. The curve in use is a round description of published ranges and it is not any measured hall. The dynamic of a passage is also a single figure for a whole audience, which is the sort of simplification a real one would not recognise.

Nothing here is heard as a texture. The excitation pattern measures loudness alone. A tail a third of a second old from a note that is no longer playing is also a harmonic presence: it colours the last chord and makes a two-note sonority of it, and a listener who hears that E5 hanging over the G3 hears an interval as much as a level. Whether that is pleasant, and whether composers aim at it, is not touched by a loudness total. The phrase the ghost of the one before is an image, and the ear’s own handling of a decaying tail, the very thing a chord is not as loud as its notes begins to address, is left out.

And the bars are all two seconds long. The last-leaver result rests on the ratio of a bar to the reverberation time. At a much slower tempo a two-second hall would forget even the last leaver, and a fast thinning would remember several. That is the same arithmetic and a single further sweep, and it may well matter on a passage whose texture falls in a matter of beats and not bars.

Whose endings these are #

The evidence is about a sustained texture of six string-like parts dropping out one a bar in a constant dynamic: the sort of ending found in slow movements of the orchestral and choral repertoire of the eighteenth and nineteenth centuries, and the endings that the closure essays have been treating as a reference. Most endings are not that. Many thin while the remaining parts get softer as well, and a part that leaves is not usually a chord tone that goes silent between bars. Any ending with a final rest or a stroke of timpani is a different case. Nothing here reaches past the stated scoring to what any orchestrator did or intended.

Still open: what the last two notes could be #

If the arrival of a thinned ending is a two-note matter, then the composer’s freedom is much larger than the spacing of a single chord, and it can be searched. Take any pair of parts as the last leaver and the final note, in every register, with their timbres and at their own levels, and ask two questions of the same model: which pair keeps the deepest arrival in a given room, and which pair keeps the arrival nearly the same across rooms, so that a score travels? A pair a minor third apart in the low register may be the most robust thing on the keyboard, since its upper note would mostly lie under the lower’s own skirt. The computation would say whether there is a best last interval for a thinning to close on, and whether that interval depends on the hall or holds in all of them.